If I use the Mate's ability to swap my face-down card with another player, can I tell another player, with a Pirate, the value of the card I swapped, so they can eliminate that other player? Is that within the spirit of the Table Talk rules?

That is exactly the spirit of the table talk rules. Stab your buddies and sell them out for your own gain. That is what any good pirate would do.

Franklin uses his Mate to swap his face down card with Amanda's. Franklin tells Leo (who has a Pirate) that Amanda's face down card is a 5. Leo uses his Pirate on his turn to eliminate Amanda (and knock).Amanda is out of this round, but is still allowed to table talk.A good pirate always gets revenge. Amanda tells Leo that Franklin's face down card is a 7.Franklin is now in deep trouble . . .